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In mathematical logic, a proof calculus corresponds to a family of formal systems that use a common style of formal inference for its inference rules. The specific inference rules of a member of such a family characterize the theory of a logic. Usually a given proof calculus encompasses more than a single particular formal system, since many proof calculi are under-determining and can be used for radically different logics. For example, a paradigmatic case is the sequent calculus, which can be used to express the consequence relations of both intuitionistic logic and relevance logic. Thus, loosely speaking, a proof calculus is a template or design pattern, characterized by a certain style of formal inference, that may be specialized to produce specific formal systems, namely by specifying the actual inference rules for such a system. There is no consensus among logicians on how best to define the term. ==Examples of proof calculi== The most widely known proof calculi are those classical calculi that are still in widespread use: *The class of Hilbert systems, of which the most famous example is the 1928 Hilbert-Ackermann system of first-order logic; *Gerhard Gentzen's calculus of natural deduction, which is the first formalism of structural proof theory, and which is the cornerstone of the formulae-as-types correspondence relating logic to functional programming; *Gentzen's sequent calculus, which is the most studied formalism of structural proof theory. Many other proof calculi were, or might have been, seminal, but are not widely used today. *Aristotle's syllogistic calculus, presented in the ''Organon'', readily admits formalisation. There is still some modern interest in syllogistic, carried out under the aegis of term logic. *Gottlob Frege's two-dimensional notation of the ''Begriffsschrift'' is usually regarded as introducing the modern concept of quantifier to logic. *C.S. Peirce's existential graph might easily have been seminal, had history worked out differently. Modern research in logic teems with rival proof calculi: *Several systems have been proposed which replace the usual textual syntax with some graphical syntax. Proof nets and cirquent calculus are among such systems. *Recently, many logicians interested in structural proof theory have proposed calculi with deep inference, for instance display logic, hypersequents, the calculus of structures, and bunched implication. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Proof calculus」の詳細全文を読む スポンサード リンク
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